The club earned a 16th-place finish following promotion last season before earning a place in Europe for the first time in their history after reaching the FA Cup final
Steve Bruce's men achieved a 16th-place finish after returning to the top flight last term, and also brought European football to the club for the first time in their history by virtue of reaching the FA Cup final.
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"Not too many pundits gave us a chance last season. Even a third of the way through the season they didn't give us a chance," he told The Hull Daily Mail.
"It was great to prove a fair few people wrong and hopefully we can do that again this season. There's a bit of depth in our squad now and that's important.
"We've got a couple from last season back to fitness now, like Sone Aluko and Robbie Brady, and that's like having two new signings."
Livermore attributes some of Hull's progress to a strong British core, with the Robert Snodgrass, Tom Ince, Harry Maguire and Andy Robertson joining the likes of Tom Huddlestone, James Chester and Curtis Davies.
"You don't see it too often in the Premier League and it's a massive part of our togetherness," he added. "We've got a great young squad, a British squad, and it's great to be a part of it.
"We're like a family; we travel away together and work hard for each other. The manager has signed players who he feels are willing and capable of joining in that."